I have retained 3; then, saying 5 times 3 are 15, and the 3 retained are 18, I see that 5 was too great, since 18 cannot be taken from 17, I have tried 4, which I have found right; I have then written it to the quotient; multiplying and subtracting, I have 248 for remainder; taking down the 9, I have had 2489 for my last partial dividend, which has given 6 for quotient, with the remainder 281, which I set down by the quotient, under the form of a division not executed, and which in reality can only be indicated, since a farther division is impossible.

N. B. Care must be taken to put a dot under each of the figures of the dividend, as you are taking them down to form the partial dividends, in order to avoid the oversight of taking down the same twice; these dots will besides indicate the number of figures which must be had in the quotient.

REMARK 1st. Division can always be shortened, when the divisor is terminated by a number of cyphers; with regard to the dividend, either it has a number of cyphers, equal or superior to that in the divisor, or it has none at all, or it has a smaller number.

548000 700

In the 1st. case, you Example.-First Case. must strike off, in both members, an equal number of cyphers; which, as it is evident, will give the same quotient; for, by cutting off one, two, three, &c. cyphers, at the right of

write

5480

12

the divisor, you render this number 10, 100, 1000 &c. times smaller, and therefore capable of being contained that number of times more in the dividend: the quotient is then restored, if, at the same time: you make the dividend proportionably smaller.

The way to shorten division in the 2d. case, is to

Example. Second Case.

674 345

separate as many figures on
the left of the dividend, as
there are cyphers on the 48
right of the divisor, which
are also to be separated by
a line; then make your
division with the two mem-
bers thus reduced; but the
operation being done, you

194
192.

2345

24 (000

28 2345

24000

must take down, to the right of the remainder, the figures which had been severed at the right of the dividend..

In the present example we have followed the process indicated above, of which this is the explication; it is evident from what has been said on the first case, that taking 674 for dividend, and 24 as divisor, we have acted the same as if the total dividend had been 674000, and the divisor 24000; but as the number 345 is a portion of the given dividend, this portion must necessarily be added to the remainder 2, which, being a number of thousands by the rank it holds, the total remainder will be 2345; the exact quotient will then be, 28; for, since, with respect to the remainder of the division, we reinstate the dividend in its integrity, we must do the same with the divisor.

REMARK 2nd. The quotient of a division expressing the number of times the divisor is contained in the dividend, it is plain that the product of the divisor by the quotient, must give the dividend; thus, if I have 24 to divide by 4, the question is to find a number, which, multiplied by 4, be equal to 24; this number must then be 4 times smaller than 24, or the fourth of 24; whence it follows that to divide a

number by 2, 3, 4,.....10, &c. is the same as to take the half, the third, the fourth.....the tenth part of that number.

DIVISION OF DECIMALS.

The division of numbers which contain decimal parts, can always be executed without difficulty by using a general method, which consists in having the same number of decimals in both members, which will be obtained by adding cyphers at the right of the member which contains fewer decimals; then omitting the comma, make the division, which will give the quotient asked for for, it is evident, that rendering, by the suppression of the comma, both dividend and divisor the same number of times greater, we change nothing in the quotient.

EXAMPLE. Be it proposed to divide 34,7 by 3,506; we must write 34700 3506

It sometimes happens that the dividend is not exactly divisible by the divisor, and then the true quotient is to be obtained only within a unit; yet, in many cases, it is necessary to obtain a more exact quotient; the decimals furnish us with an easy mean of obtaining it within as approximate a degree as can be wished, viz. within a tenth, a hundredth, a thousandth, &c. we need for that but add as many cyphers at the right of the dividend as we wish to have decimals in the quotient.

EXAMPLE.

Be it asked for the quotient of 459 by 23, within a hundredth part of unit.

Since we must have two decimals in the quotient, we must then make the dividend a hundred times

too great, by writing two cyphers to the right of that number, and the division to be executed will

be

That total quotient 1995 which has been obtained, is a hundred times too great, since we had made the dividend a hundred times too great; the quotient must then be rendered a hundred times smaller, which, as has already been said in Numeration, is effected by separating with a comma two figures on the right of that number, and it

gives for the true quotient, nineteen units and ninetyfive hundredths; here it is evident that the quotient is right within one hundredth, for, if instead of 5 hundredths, we had written 6, this figure would be too high.

PROOF OF DIVISION AND MULTIPLICATION.

When a division is done right, the product of the divisor by the quotient, added to the remainder, if there be any, must be equal to the dividend. Thus, the proof of division is done by multiplication, and consequently multiplication is also proved by division; for, dividing the product of a multiplication by one of its factors, the question must be the other factor.

OF FRACTIONS.

A fraction is a quantity less than a unit; if then we consider the unit divided into two, three, four,

kc. equal parts, the half, the two thirds, the three fourths, of the principal unit will be fractions.

Two numbers are necessary to express a fraction, they are written one under the other, and separated by a line, just as a division when it is only indicated; thus represent a fraction, which we read, three fourths; it means the three fourths of a unit. The superior number is called numerator, and the inferior denominator. This one shews into how many equal parts the principal unit is supposed to be divided, and the numerator expresses how many of these parts are taken.

This is the most direct manner of considering a fraction; but there is another not less important, as it easily accounts for several operations to be performed on fractions: it is to consider the two terms of a fraction as the two members of a division, whereof the numerator is the dividend, and the denominator the divisor. But we must be fully convinced that this new manner of considering a fraction, leads to the same result as the former: let us take the fraction for example; we must prove, that, dividing the unit into four equal parts, and taking three of them, we shall have the same result, as if taking three of the same units, we divided them by four. This proof can be rendered sensible by an application.

Let us suppose that the principal unit be the pound, which is worth 20 shillings: from the first manner of considering a fraction, we must divide the pound into four equal parts, each of which will be 5s. and then taking three of these parts, we shall have 15s. Now, if 3, represent a dividend, it must be considered as 3 units, or 3£. or 60s. but 60s, divided by 4, give 15s. we have then the same quotient as above; we may, therefore, indifferently consider fractions in either of

these two manners.